Journal articles: 'Complex Differential Geometry'

1

Beggs, Edwin, and S. Paul Smith. "Non-commutative complex differential geometry." Journal of Geometry and Physics 72 (October 2013): 7–33. http://dx.doi.org/10.1016/j.geomphys.2013.03.018.

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Wang, Shuguang. "Twisted complex geometry." Journal of the Australian Mathematical Society 80, no. 2 (April 2006): 273–96. http://dx.doi.org/10.1017/s1446788700013112.

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AbstractWe introduce complex differential geometry twisted by a real line bundle. This provides a new approach to understand the various real objects that are associated with an anti-holomorphic involution. We also generalize a result of Greenleaf about real analytic sheaves from dimension 2 to higher dimensions.

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Donaldson, S. "DIFFERENTIAL GEOMETRY OF COMPLEX VECTOR BUNDLES." Bulletin of the London Mathematical Society 21, no. 1 (January 1989): 104–6. http://dx.doi.org/10.1112/blms/21.1.104.

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Donaldson, S. K. "Some Numerical Results in Complex Differential Geometry." Pure and Applied Mathematics Quarterly 5, no. 2 (2009): 571–618. http://dx.doi.org/10.4310/pamq.2009.v5.n2.a2.

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McKay, B. "Complex nonlinear ordinary differential equations and geometry." Journal of Physics: Conference Series 55 (December 1, 2006): 165–70. http://dx.doi.org/10.1088/1742-6596/55/1/016.

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Anco, Stephen, John Bland, and Michael Eastwood. "Some Penrose transforms in complex differential geometry." Science in China Series A: Mathematics 49, no. 11 (November 2006): 1599–610. http://dx.doi.org/10.1007/s11425-006-2066-5.

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Okonek, Christian. "Book Review: Differential geometry of complex vector bundles." Bulletin of the American Mathematical Society 19, no. 2 (October 1, 1988): 528–31. http://dx.doi.org/10.1090/s0273-0979-1988-15731-x.

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Muñoz Velázquez, Vicente. "The Hodge conjecture: The complications of understanding the shape of geometric spaces." Mètode Revista de difusió de la investigació, no. 8 (June 5, 2018): 51. http://dx.doi.org/10.7203/metode.0.8253.

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The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. It was proposed by William Hodge in 1950 and is currently a stimulus for the development of several theories based on geometry, analysis, and mathematical physics. It proposes a natural condition for the existence of complex submanifolds within a complex manifold. Manifolds are the spaces in which geometric objects can be considered. In complex manifolds, the structure of the space is based on complex numbers, instead of the most intuitive structure of geometry, based on real numbers.

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Everitt, W. N., and L. Markus. "Complex symplectic geometry with applications to ordinary differential operators." Transactions of the American Mathematical Society 351, no. 12 (July 20, 1999): 4905–45. http://dx.doi.org/10.1090/s0002-9947-99-02418-6.

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Aleksandrov, A. G. "Residues of Logarithmic Differential Forms in Complex Analysis and Geometry." Analysis in Theory and Applications 30, no. 1 (June 2014): 34–50. http://dx.doi.org/10.4208/ata.2014.v30.n1.3.

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Syed Khawar Nadeem Kirmani and Raja Noshad Jamil. "Optimization of Complex Geometry Using Tenth Order Partial Differential Equation." Scientific Inquiry and Review 2, no. 2 (April 30, 2018): 23–31. http://dx.doi.org/10.32350/sir/22/020203.

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This paper presents an efficient and intuitive technique of shape parameterization for design optimization using a partial differential equation (PDE) of order ten. It shows how the choice of two introduced parameters can enable one to parameterize complex geometries. With the use of PDE based formulation, it is shown in this paper how the shape can be defined and manipulated on the basis of parameterization and the boundary value approaches by which complex shapes can be created. Further the boundary conditions which are used in it are a boundary and an intermediate curves for defining the shape. This technique allows complex shapes to be parameterized intuitively using a very small set of design parameters. Hence, Practical design optimization of problems becomes more achievable by applying PDE based approach of shape parameterization by incorporating standard numerical optimization techniques.

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Karimani, S. K. N., and R. N. Jamil. "Optimization of Complex Geometry Using Tenth Order Partial Differential Equation." Scientific Inquiry and Review 2, no. 2 (April 2018): 22–27. http://dx.doi.org/10.29145/sir/22/020203.

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Beardon, A. F. "DIFFERENTIAL GEOMETRY AND COMPLEX ANALYSIS H. E. RAUCH MEMORIAL VOLUME." Bulletin of the London Mathematical Society 18, no. 4 (July 1986): 427. http://dx.doi.org/10.1112/blms/18.4.427a.

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Liu, Kefeng, and Xiaonan Ma. "A remark on 'Some numerical results in complex differential geometry'." Mathematical Research Letters 14, no. 2 (2007): 165–71. http://dx.doi.org/10.4310/mrl.2007.v14.n2.a1.

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Baldi, Annalisa, Maria Carla Tesi, and Francesca Tripaldi. "Sobolev-Gaffney type inequalities for differential forms on sub-Riemannian contact manifolds with bounded geometry." Advanced Nonlinear Studies 22, no. 1 (January 1, 2022): 484–516. http://dx.doi.org/10.1515/ans-2022-0022.

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Abstract In this article, we establish a Gaffney type inequality, in W ℓ , p {W}^{\ell ,p} -Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind noncompact manifolds). Here, p ∈ ] 1 , ∞ [ p\in ]1,\infty {[} and ℓ = 1 , 2 \ell =1,2 depending on the order of the differential form we are considering. The proof relies on the structure of the Rumin’s complex of differential forms in contact manifolds, on a Sobolev-Gaffney inequality proved by Baldi-Franchi in the setting of the Heisenberg groups and on some geometric properties that can be proved for sub-Riemannian contact manifolds with bounded geometry.

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Ali, Sajid, M. Safdar, and Asghar Qadir. "Linearization from Complex Lie Point Transformations." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/793247.

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Complex Lie point transformations are used to linearize a class of systems of second order ordinary differential equations (ODEs) which have Lie algebras of maximum dimensiond, withd≤4. We identify such a class by employing complex structure on the manifold that defines the geometry of differential equations. Furthermore we provide a geometrical construction of the procedure adopted that provides an analogue inR3of the linearizability criteria inR2.

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Ogievetskii, O. V. "Complex differential geometry and supermanifolds in theories of strings and fields." Uspekhi Fizicheskih Nauk 159, no. 12 (1989): 722. http://dx.doi.org/10.3367/ufnr.0159.198912g.0722.

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Qadir, Asghar. "Linearization: Geometric, Complex, and Conditional." Journal of Applied Mathematics 2012 (2012): 1–30. http://dx.doi.org/10.1155/2012/303960.

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Lie symmetry analysis provides a systematic method of obtaining exact solutions of nonlinear (systems of) differential equations, whether partial or ordinary. Of special interest is the procedure that Lie developed to transform scalar nonlinear second-order ordinary differential equations to linear form. Not much work was done in this direction to start with, but recently there have been various developments. Here, first the original work of Lie (and the early developments on it), and then more recent developments based on geometry and complex analysis, apart from Lie’s own method of algebra (namely, Lie group theory), are reviewed. It is relevant to mention that much of the work isnotlinearization but uses the base of linearization.

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Chen, Bang-Yen, Adara M. Blaga, and Gabriel-Eduard Vîlcu. "Differential Geometry of Submanifolds in Complex Space Forms Involving δ-Invariants." Mathematics 10, no. 4 (February 14, 2022): 591. http://dx.doi.org/10.3390/math10040591.

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One of the fundamental problems in the theory of submanifolds is to establish optimal relationships between intrinsic and extrinsic invariants for submanifolds. In order to establish such relations, the first author introduced in the 1990s the notion of δ-invariants for Riemannian manifolds, which are different in nature from the classical curvature invariants. The earlier results on δ-invariants and their applications have been summarized in the first author’s book published in 2011 Pseudo-Riemannian Geometry, δ-Invariants and Applications (ISBN: 978-981-4329-63-7). In this survey, we present a comprehensive account of the development of the differential geometry of submanifolds in complex space forms involving the δ-invariants done mostly after the publication of the book.

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Ogievetskiĭ, O. V. "Complex differential geometry and supermanifolds in theories of strings and fields." Soviet Physics Uspekhi 32, no. 12 (December 31, 1989): 1116. http://dx.doi.org/10.1070/pu1989v032n12abeh002788.

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Yılmaz, Süha. "Contributions to differential geometry of isotropic curves in the complex space." Journal of Mathematical Analysis and Applications 374, no. 2 (February 2011): 673–80. http://dx.doi.org/10.1016/j.jmaa.2010.09.031.

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Mercati, Flavio. "Quantum κ-deformed differential geometry and field theory." International Journal of Modern Physics D 25, no. 05 (April 2016): 1650053. http://dx.doi.org/10.1142/s021827181650053x.

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I introduce in [Formula: see text]-Minkowski noncommutative spacetime the basic tools of quantum differential geometry, namely bicovariant differential calculus, Lie and inner derivatives, the integral, the Hodge-[Formula: see text] and the metric. I show the relevance of these tools for field theory with an application to complex scalar field, for which I am able to identify a vector-valued four-form which generalizes the energy–momentum tensor. Its closedness is proved, expressing in a covariant form the conservation of energy–momentum.

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Richter, Wolf-Dieter. "On Hyperbolic Complex Numbers." Applied Sciences 12, no. 12 (June 8, 2022): 5844. http://dx.doi.org/10.3390/app12125844.

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For dimensions two, three and four, we derive hyperbolic complex algebraic structures on the basis of suitably defined vector products and powers which allow in a standard way a series definitions of the hyperbolic vector exponential function. In doing so, we both modify arrow multiplication, which, according to Feynman, is fundamental for quantum electrodynamics, and we give a geometric explanation of why in a certain situation it is natural to define random vector products. Through the interplay of vector algebra, geometry and complex analysis, we extend a systematic approach previously developed for various other complex algebraic structures to the field of hyperbolic complex numbers. We discuss a quadratic vector equation and the property of hyperbolically holomorphic functions of satisfying hyperbolically modified Cauchy–Riemann differential equations and also give a proof of an Euler type formula based on series expansion.

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Chen, Changjun, Rishu Saxena, and Guo-Wei Wei. "A Multiscale Model for Virus Capsid Dynamics." International Journal of Biomedical Imaging 2010 (2010): 1–9. http://dx.doi.org/10.1155/2010/308627.

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Viruses are infectious agents that can cause epidemics and pandemics. The understanding of virus formation, evolution, stability, and interaction with host cells is of great importance to the scientific community and public health. Typically, a virus complex in association with its aquatic environment poses a fabulous challenge to theoretical description and prediction. In this work, we propose a differential geometry-based multiscale paradigm to model complex biomolecule systems. In our approach, the differential geometry theory of surfaces and geometric measure theory are employed as a natural means to couple the macroscopic continuum domain of the fluid mechanical description of the aquatic environment from the microscopic discrete domain of the atomistic description of the biomolecule. A multiscale action functional is constructed as a unified framework to derive the governing equations for the dynamics of different scales. We show that the classical Navier-Stokes equation for the fluid dynamics and Newton's equation for the molecular dynamics can be derived from the least action principle. These equations are coupled through the continuum-discrete interface whose dynamics is governed by potential driven geometric flows.

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Arnold, Douglas N., Richard S. Falk, and Ragnar Winther. "Finite element exterior calculus, homological techniques, and applications." Acta Numerica 15 (May 2006): 1–155. http://dx.doi.org/10.1017/s0962492906210018.

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Finite element exterior calculus is an approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are revealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Laplacian, Maxwell’s equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.

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Ming, Yang, and Lin Mian. "A Novel Method for Analyzing Pore Size Distribution of Complex Geometry Shaped Porous Shale." Materials Science Forum 1003 (July 2020): 134–43. http://dx.doi.org/10.4028/www.scientific.net/msf.1003.134.

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This article proposes the differential BJH equation based on the principles of multilayer adsorption and capillary condensation, which was simplified by theoretical investigation and experiments. This work indicates that the differential function of isotherm and the differential function of pore size to relative pressure determine the pore size distribution of porous media. The differential BJH model can be used to explain the source of the false peak in pore size distribution and to calculate the pore size distribution of different shapes of pores in a porous media with a porous structure. It has an excellent application prospect in the characterization of complex pore structure represented by shale.

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Chu, Chong-Sun, and Pei-Ming Ho. "Poisson Algebra of Differential Forms." International Journal of Modern Physics A 12, no. 31 (December 20, 1997): 5573–87. http://dx.doi.org/10.1142/s0217751x97002929.

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We give a natural definition of a Poisson differential algebra. Consistency conditions are formulated in geometrical terms. It is found that one can often locally put the Poisson structure on the differential calculus in a simple canonical form by a coordinate trans-formation. This is in analogy with the standard Darboux's theorem for symplectic geometry. For certain cases there exists a realization of the exterior derivative through a certain canonical one-form. All the above are carried out similarly for the case of a complex Poisson differential algebra. The case of one complex dimension is treated in detail and interesting features are noted. Conclusions are made in the last section.

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El-Nabulsi, Rami Ahmad. "Higher-Order Geodesic Equations from Non-Local Lagrangians and Complex Backward-Forward Derivative Operators." Annals of West University of Timisoara - Mathematics and Computer Science 54, no. 1 (July 1, 2016): 139–57. http://dx.doi.org/10.1515/awutm-2016-0008.

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Abstract Starting with an extended complex backwardforward derivative operator in differential geometry which is motivated from non-local-in-time Lagrangian dynamics, higher-order geodesic equations are obtained within classical differential geometrical settings. We limit our analysis up to the 2nd-order derivative where some applications are discussed and a number of features are revealed accordingly.

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Schetelig, B., J. Keghie, R. Kanyou Nana, L. O. Fichte, S. Potthast, and S. Dickmann. "Simplified modeling of EM field coupling to complex cable bundles." Advances in Radio Science 8 (October 1, 2010): 211–17. http://dx.doi.org/10.5194/ars-8-211-2010.

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Abstract. In this contribution, the procedure "Equivalent Cable Bundle Method" is used for the simplification of large cable bundles, and it is extended to the application on differential signal lines. The main focus is on the reduction of twisted-pair cables. Furthermore, the process presented here allows to take into account cables with wires that are situated quite close to each other. The procedure is based on a new approach to calculate the geometry of the simplified cable and uses the fact that the line parameters do not uniquely correspond to a certain geometry. For this reason, an optimization algorithm is applied.

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PAYCHA, SYLVIE. "RENORMALIZED TRACES AS A LOOKING GLASS INTO INFINITE-DIMENSIONAL GEOMETRY." Infinite Dimensional Analysis, Quantum Probability and Related Topics 04, no. 02 (June 2001): 221–66. http://dx.doi.org/10.1142/s0219025701000486.

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This paper, based on results obtained in recent years with various coauthors,1–3,13,53 presents a proposal to extend some classical geometric concepts to a class of infinite-dimensional manifolds such as current groups and to a class of infinite-dimensional bundles including the ones arising in the family index theorem. The basic idea is to extend the notion of trace underlying many geometric concepts using renormalized traces which are linear functionals on pseudo-differential operators. The definition of "renormalized traces" involves extra data on the manifolds or vector bundles, namely a weight given by a field of elliptic operators which becomes part of the geometric data, leading to the notion of weighted manifold and weighted vector bundle. This weight is a source of anomaly arising typically as a Wodzicki residue of some pseudo-differential operator. We investigate the anomalies that arise when trying to extend to the infinite-dimensional setting classical results of finite-dimensional geometry such as a Weitzenböck formula, Chern–Weil invariants or the relation between the first Chern form on a complex vector bundle and the curvature on the associated determinant bundle. When comparable, we relate our approach to the one adopted for similar problems in noncommutative geometry.

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Phan, Minh Son, Katherine Matho, Emmanuel Beaurepaire, Jean Livet, and Anatole Chessel. "nAdder: A scale-space approach for the 3D analysis of neuronal traces." PLOS Computational Biology 18, no. 7 (July 5, 2022): e1010211. http://dx.doi.org/10.1371/journal.pcbi.1010211.

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Tridimensional microscopy and algorithms for automated segmentation and tracing are revolutionizing neuroscience through the generation of growing libraries of neuron reconstructions. Innovative computational methods are needed to analyze these neuronal traces. In particular, means to characterize the geometric properties of traced neurites along their trajectory have been lacking. Here, we propose a local tridimensional (3D) scale metric derived from differential geometry, measuring for each point of a curve the characteristic length where it is fully 3D as opposed to being embedded in a 2D plane or 1D line. The larger this metric is and the more complex the local 3D loops and turns of the curve are. Available through the GeNePy3D open-source Python quantitative geometry library (https://genepy3d.gitlab.io), this approach termed nAdder offers new means of describing and comparing axonal and dendritic arbors. We validate this metric on simulated and real traces. By reanalysing a published zebrafish larva whole brain dataset, we show its ability to characterize different population of commissural axons, distinguish afferent connections to a target region and differentiate portions of axons and dendrites according to their behavior, shedding new light on the stereotypical nature of neurites’ local geometry.

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Yılmaz, Süha, and Yasin Ünlütürk. "Contributions to differential geometry of isotropic curves in the complex space C3 – II." Journal of Mathematical Analysis and Applications 440, no. 2 (August 2016): 561–77. http://dx.doi.org/10.1016/j.jmaa.2016.02.072.

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Cotti, Giordano, and Davide Guzzetti. "Analytic geometry of semisimple coalescent Frobenius structures." Random Matrices: Theory and Applications 06, no. 04 (October 2017): 1740004. http://dx.doi.org/10.1142/s2010326317400044.

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We present some results of a joint paper with Dubrovin (see references), as exposed at the Workshop “Asymptotic and Computational Aspects of Complex Differential Equations” at the CRM in Pisa, in February 2017. The analytical description of semisimple Frobenius manifolds is extended at semisimple coalescence points, namely points with some coalescing canonical coordinates although the corresponding Frobenius algebra is semisimple. After summarizing and revisiting the theory of the monodromy local invariants of semisimple Frobenius manifolds, as introduced by Dubrovin, it is shown how the definition of monodromy data can be extended also at semisimple coalescence points. Furthermore, a local Isomonodromy theorem at semisimple coalescence points is presented. Some examples of computation are taken from the quantum cohomologies of complex Grassmannians.

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BAKANOV, G. B., and S. К. MELDEBEKOVA. "STABILITY OF THE DIFFERENTIAL-DIFFERENCE ANALOG OF THE INTEGRAL GEOMETRY PROBLEM WITH A WEIGHT FUNCTION." Q A Iasaýı atyndaǵy Halyqaralyq qazaq-túrіk ýnıversıtetіnіń habarlary (fızıka matematıka ınformatıka serııasy), no. 1 (March 15, 2022): 67–75. http://dx.doi.org/10.47526/2022-2/2524-0080.06.

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In this paper, we consider the problem of Integral geometry, which is brought to the problem of difference for a mixed-type equation for a bunch of curves that satisfy some regularity conditions. The study of distinctive analogues of Integral geometry problems has its own complex points, due to the fact that for limited-distinctive analogues of independent derivatives, the main relations are carried out with a certain shift over a discrete variable. Therefore, many relationships obtained in continuous representation take on a more complex form when switching to a discrete analog, and require further research on the connectors that occur duringthe shift. Since these problems do not have a theorem on the existence of a solution in the general case, the concept of conditional correctness is used, that is, it is assumed that the problem of Integral geometry and its differential-differential analoghave a solution. The stability assessment of the differential analog of the boundary problem for the mixed-type equation obtained in the work is carried out by geotomography, medical tomography, defectoscopy, etc. it is used to justify the compactness of numerical problem solving methods.

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Akitsu, Takashiro. "Mathematical Geometry and Groups for Low-Symmetry Metal Complex Systems." Molecules 28, no. 11 (June 2, 2023): 4509. http://dx.doi.org/10.3390/molecules28114509.

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Since chemistry, materials science, and crystallography deal with three-dimensional structures, they use mathematics such as geometry and symmetry. In recent years, the application of topology and mathematics to material design has yielded remarkable results. It can also be seen that differential geometry has been applied to various fields of chemistry for a relatively long time. There is also the possibility of using new mathematics, such as the crystal structure database, which represents big data, for computational chemistry (Hirshfeld surface analysis). On the other hand, group theory (space group and point group) is useful for crystal structures, including determining their electronic properties and the symmetries of molecules with relatively high symmetry. However, these strengths are not exhibited in the low-symmetry molecules that are actually handled. A new use of mathematics for chemical research is required that is suitable for the age when computational chemistry and artificial intelligence can be used.

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Oluwayemi, Matthew Olanrewaju, Esther O. Davids, and Adriana Cătaş. "On Geometric Properties of a Certain Analytic Function with Negative Coefficients." Fractal and Fractional 6, no. 3 (March 21, 2022): 172. http://dx.doi.org/10.3390/fractalfract6030172.

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Various function theorists have successfully defined and investigated different kinds of analytic functions. The applications of such functions have played significant roles in geometry function theory as a field of complex analysis. In this work, therefore, a certain subclass of univalent analytic functions of the form f(z)=z−∑m=2t[ω(2+β)+cγ−σ]Cm[mσ−cω(2+β)+cγ]Knzm−∑k=t+1∞akzk is defined using a generalized differential operator. Furthermore, some geometric properties for the class were established.

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Saviuc, Alexandra, Manuela Gîrțu, Liliana Topliceanu, Tudor-Cristian Petrescu, and Maricel Agop. "“Holographic Implementations” in the Complex Fluid Dynamics through a Fractal Paradigm." Mathematics 9, no. 18 (September 16, 2021): 2273. http://dx.doi.org/10.3390/math9182273.

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Assimilating a complex fluid with a fractal object, non-differentiable behaviors in its dynamics are analyzed. Complex fluid dynamics in the form of hydrodynamic-type fractal regimes imply “holographic implementations” through velocity fields at non-differentiable scale resolution, via fractal solitons, fractal solitons–fractal kinks, and fractal minimal vortices. Complex fluid dynamics in the form of Schrödinger type fractal regimes imply “holographic implementations”, through the formalism of Airy functions of fractal type. Then, the in-phase coherence of the dynamics of the complex fluid structural units induces various operational procedures in the description of such dynamics: special cubics with SL(2R)-type group invariance, special differential geometry of Riemann type associated to such cubics, special apolar transport of cubics, special harmonic mapping principle, etc. In such a manner, a possible scenario toward chaos (a period-doubling scenario), without concluding in chaos (nonmanifest chaos), can be mimed.

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Szalay, Tibor, Mátyás Horváth, and Balázs Tukora. "Hybrid Experimental-Dynamical Modelling of Cutting Considering the Complex Chip Geometry." Applied Mechanics and Materials 88-89 (August 2011): 741–45. http://dx.doi.org/10.4028/www.scientific.net/amm.88-89.741.

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The introduced cutting model uses the widely applied mechanical-dynamical differential Lagrange model together with one of the most popular experimental force model (Kienzle-Victor model) in order to provide the more accurate demonstration and simulation of cutting processes. To increase the reliability of the results the authors considered as much parameters and as complex chip geometry as the calculations and processing made it possible. In this paper the sophisticated model of milling operation was the aim of the authors. The simulation results show good equivalency with the measured real cutting experiments. In spite of the complexity of the equations the rapid development in the informatics (hardware and software tools) helped the handling and quick calculation of the equations in this type of models.

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Ionescu, Lucian-Miti, Cristina-Liliana Pripoae, and Gabriel-Teodor Pripoae. "Classification of Holomorphic Functions as Pólya Vector Fields via Differential Geometry." Mathematics 9, no. 16 (August 9, 2021): 1890. http://dx.doi.org/10.3390/math9161890.

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We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f(z)=b(z+d)−1. We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields.

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Wu, Hao, Yongqiang Cheng, Xiaoqiang Hua, and Hongqiang Wang. "Vector Bundle Model of Complex Electromagnetic Space and Change Detection." Entropy 21, no. 1 (December 23, 2018): 10. http://dx.doi.org/10.3390/e21010010.

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Complex Electromagnetic Space (CEMS), which consists of physical space and the complex electromagnetic environment, plays an essential role in our daily life for supporting remote communication, wireless network, wide-range broadcast, etc. In CEMS, the electromagnetic activities might work differently from the ideal situation; the typical case is that undesired signal would disturb the echo of objects and overlap into it resulting in the mismatch of matched filter and the reduction of the probability of detection. The lacking mathematical description of CEMS resulting from the complexity of electromagnetic environment leads to the inappropriate design of detection method. Therefore, a mathematical model of CEMS is desired for integrating the electromagnetic signal in CEMS as a whole and considering the issues in CEMS accurately. This paper puts forward a geometric model of CEMS based on vector bundle, which is an abstract concept in differential geometry and proposes a geometric detector for change detection in CEMS under the geometric model. In the simulation, the proposed geometric detector was compared with energy detector and matched filter in two scenes: passive detection case and active detection case. The results show the proposed geometric detector is better than both energy detector and matched filter with 4∼5 dB improvements of SNR (signal-to-noise ratio) in two scenes.

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Erdal, Ibrahim, and Ekin Uğurlu. "Second-Order Multiparameter Problems Containing Complex Potentials." Axioms 11, no. 12 (December 8, 2022): 706. http://dx.doi.org/10.3390/axioms11120706.

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In this work, we provide some lower bounds for the number of squarly integrable solutions of some second-order multiparameter differential equations. To obtain the results, we use both Sims and Sleeman’s ideas and the results are some generalization of the known results. To be more precise, we firstly construct the Weyl–Sims theory for the singular second-order differential equation with several spectral parameters. Then, we obtain some results for the several singular second-order differential equations with several spectral parameters.

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BRZEZIŃSKI, TOMASZ. "NON-COMMUTATIVE CONNECTIONS OF THE SECOND KIND." Journal of Algebra and Its Applications 07, no. 05 (October 2008): 557–73. http://dx.doi.org/10.1142/s0219498808002977.

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A connection-like objects, termed hom-connections are defined in the realm of non-commutative geometry. The definition is based on the use of homomorphisms rather than tensor products. It is shown that hom-connections arise naturally from (strong) connections in non-commutative principal bundles. The induction procedure of hom-connections via a map of differential graded algebras or a differentiable bimodule is described. The curvature for a hom-connection is defined, and it is shown that flat hom-connections give rise to a chain complex.

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Kamiya, Noriaki, and Susumu Okubo. "On δ-Lie supertriple systems associated with (ε, δ)-Freudenthal-Kantor supertriple systems." Proceedings of the Edinburgh Mathematical Society 43, no. 2 (June 2000): 243–60. http://dx.doi.org/10.1017/s0013091500020903.

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AbstractWe will present an investigation of (ε, δ)-Freudenthal–Kantor supertriple systems that are intimately related to Lie supertriple systems and Lie superalgebras. We can also introduce a super analogue of Nijenhuis tensor and almost-complex structure in differential geometry.

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Manev, Mancho. "On canonical-type connections on almost contact complex Riemannian manifolds." Filomat 29, no. 3 (2015): 411–25. http://dx.doi.org/10.2298/fil1503411m.

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We consider a pair of smooth manifolds, which are the counterparts in the even-dimensional and odd-dimensional cases. They are separately an almost complex manifold with Norden metric and an almost contact manifolds with B-metric, respectively. They can be combined as the so-called almost contact complex Riemannian manifold. This paper is a survey with additions of results on differential geometry of canonical-type connections (i.e. metric connections with torsion satisfying a certain algebraic identity) on the considered manifolds.

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45

Nishioka, Keiji. "Differential algebraic function fields depending rationally on arbitrary constants." Nagoya Mathematical Journal 113 (March 1989): 173–79. http://dx.doi.org/10.1017/s0027763000001331.

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The general solution of an algebraic differential equation depends on the initial conditions, though it is in general too difficult to make explicit the shape of the relationship. Painlevé studied in [8] algebraic differentia] equations of second order with the general solutions depending rationally on the initial conditions and the solvability of such equations. Giving the precise definition of the notion “rational dependence on the initial conditions”, Umemura [10] revived and generalized rigorously the discussion of Painlevé in the language of modern algebraic geometry. The theorem of Umemura is as follows; Let K be a differential field extension of complex number field C generated by a finite number of meromorphic functions on some domain in C. Let y be the general solution of a given algebraic differential equation over K. Suppose that y depends rationally on the initial conditions. Then it is contained in the terminal Km of a finite chain of differential field extensions: K = K0 ⊂ K1 ⊂… ⊂Km such that each Ki is strongly normal over Ki−1.

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RESENDES, D. P. "Geometric algebra in plasma electrodynamics." Journal of Plasma Physics 79, no. 5 (April 12, 2013): 735–38. http://dx.doi.org/10.1017/s0022377813000366.

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AbstractGeometric algebra (GA) is a recent broad mathematical framework incorporating synthetic and coordinate geometry, complex variables, quarternions, vector analysis, matrix algebra, spinors, tensors, and differential forms. It has been claimed to be a unified language for physics. GA is presented in the context of the Maxwell-Plasma system. In this formalism the divergence and curl differential operators are united in a single vector derivative, which is invertible, in the form of a first-order Green function. The four Maxwell equations can be combined into a single equation (for homogeneous and constant media) or into two equations involving the invertible vector derivative for more complex media. GA is applied to simple examples to illustrate the compactness of the notation and coordinate-free computations.

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Fu, Jiajun. "Nonexistence of Surjective Lie Homomorphism Between Complex Special and General Linear Groups." Highlights in Science, Engineering and Technology 47 (May 11, 2023): 67–70. http://dx.doi.org/10.54097/hset.v47i.8166.

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Lie groups are very important objects in algebra, topology and geometry, as it is naturally endowed with two special structures: the algebraic structure of group and the geometric structure of differential manifold. Hence, it is quite meaningful to better know the structure of Lie groups and the actions, and the transformations between them. Two specific theorems about Lie groups and their homomorphism and Lie homomorphism are proved in this paper. First, the fact that there does not exist a surjective Lie group homomorphism between and is introduced. This is done by first having a trial of constructing Lie group surjective homomorphism but fails. Next, this result is further proved by applying topological method that to construct a covering map between and , followed by computing the fundamental group of . Finally, by applying that is a simple group, the fact that there does not exist a surjective group homomorphism between and is demonstrated, together with the statement that is a simple group.

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HUANG, XIN-BING. "NEW GEOMETRIC FORMALISM FOR GRAVITY EQUATION IN EMPTY SPACE." International Journal of Modern Physics D 14, no. 06 (June 2005): 1009–22. http://dx.doi.org/10.1142/s0218271805006572.

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In this paper, a complex daor field which can be regarded as the square root of space–time metric is proposed to represent gravity. The locally complexified geometry is set up, and the complex spin connection constructs a bridge between gravity and SU(1, 3) gauge field. Daor field equations in empty space are acquired, which are one-order differential equations and do not conflict with Einstein's gravity theory.

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Evans, C. J., M. I. G. Bloor, and M. J. Wilson. "Shape Parameterization of the Time-Dependent Geometry of the Heart for Steady Fluid Dynamical Analysis." Journal of Theoretical Medicine 3, no. 4 (2001): 221–30. http://dx.doi.org/10.1080/10273660108833077.

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A parametric model of the complex time-dependent geometry of the ventricles of the human heart is constructed. The geometry model is created by means of a boundary value approach, solving an elliptic partial differential equation to generate a representation of the inner surface of the ventricles. The technique provides a closed-form description of the geometry with the advantage that the geometry can be readily changed without introducing holes or discontinuities in the surface. It also allows a straightforward link to analysis, facilitating the calculation of physical properties such as those relevant to fluid dynamics. As an application of this work, the geometry model is combined with commercial CFD software to analyse the blood flow in the heart. Steady-state calculations are performed at various time steps to follow the evolution of the fluid flow.

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GEROYANNIS, V. S., and I. E. SFAELOS. "NUMERICAL TREATMENT OF ROTATING NEUTRON STARS SIMULATED BY GENERAL-RELATIVISTIC POLYTROPIC MODELS: A COMPLEX-PLANE STRATEGY." International Journal of Modern Physics C 22, no. 03 (March 2011): 219–48. http://dx.doi.org/10.1142/s0129183111016269.

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We solve numerically in the complex plane all the differential equations involved in Hartle's perturbation method for computing general-relativistic polytropic models of rotating neutron stars. We give emphasis on computing quantities describing the geometry of models in rapid rotation. Compared to numerical results obtained by certain sophisticated iterative methods, we verify appreciable improvement of our results vs to those given by the classical Hartle's perturbative scheme.

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Journal articles on the topic 'Complex Differential Geometry'

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